In this work, we constructed and analyzed the effective continuum theories corresponding to the microscopic tight binding models proposed in Ref. [1]} and [2]} based on the systematic method proposed in [3]}. The nearly perfect agreement between the dispersion of the tight binding models and the dispersion of the effective continuum theories demonstrates the correctness of the constructed continuum theories and the validity of the method. We therefore envision that the experimentally measured \(\mathbf {u}_j(\mathbf {r})\)  can be plugged into our effective Hamiltonian, and the resulting energy spectra and eigenfunctions can then be used to directly compare with the scanning tunneling spectroscopy (STS) measurement of the electronic local density of states. This may pave the way for a more quantitative comparison between the theoretical predictions and the experimental results. In addition, our theory provides electron-phonon couplings as a byproduct, which are important to fully understand the role of phonons in superconductivity of TBG.
